$7\mathrm{.}13$ Time for Cryosurgery: Spherical Ball

We would like to estimate the time needed in cryosurgery where tissue is frozen using a spherical cryosurgical probe inserted into the tissue. To make things easier, you

will be guided through the setup of the governing equation. Consider pseudo$-$steady$-$state heat transfer where the tissue touching the probe is at the liquid nitrogen

temperature of $T_p=-196C.$ Consider the tissue to be initially unfrozen, but already at the constant freezing point of $T_m=-2C.$ Ignore any sensible heat involved.

Let us say the outer frozen layer increases by a value of $dr$ in time $dt.$ Write the expression for the rate of heat flow that must be released for this layer of material

to freeze. $2)$ The heat flow in step $1$ has to be conducted through the frozen layer into the cryoprobe. Write an expression for heat conduction through the frozen layer at

steady state, considering an appropriate expression for thermal resistance of a spherical shell$\mathrm{.}3)$ Equate thre expressions in step $1$ and step $2$ to set up the equation from

which we can solve for the radius of frozen tissue at time $t\mathrm{.}4)$ Integrate from radius $r_i($at time $0)$ to any radius $r$ at time $t.$ Do not try to simplify to get $r$ as a function of

$t.5)$ Instead, provide an approximate plot of $r$ as a function of time $t,$ based on your physical understanding of the problem.